Algorithm for Achieving Consensus over Conflicting Rumours


Academic Paper, 2014
6 Pages, Grade: bac+4

Free online reading

Algorithm for Achieving Consensus Over Conflicting
Rumors: Convergence Analysis and Applications
Ismail Elouafiq, Student Member, IEEE, Amine Semma, Student Member, IEEE,
Abstract--Motivated by the large expansion in the study of
social networks, this paper deals with the problem of multiple
messages spreading over the same network using gossip
algorithms. Given two messages distributed over some nodes of
the graph, we first investigate the final distribution of the
messages given an initial state. Then, an algorithm is presented to
achieve consensus over one of the messages. Finally, a game
theoretical application and an analogy with word-of-mouth
marketing are outlined.
Index Terms--Gossip algorithms, consensus, social networks,
game theory, word-of-mouth.
I. I
NTRODUCTION
In recent years, large decentralized distributed systems such
as sensor and wireless networks require the design of
communication schemes that satisfy scalability, robustness and
graceful degradation. Consequently, information dissemination
and message spreading algorithms have generated a huge
interest.
The problem of broadcasting may be defined as spreading
some information over a graph that is unknown to its nodes.
Gossip algorithms are a set of algorithms that are used to solve
such problems. Randomized gossip is one of the most widely
used forms of gossip that have gained prominence for the
simplicity of its protocol. In a random gossip based algorithm,
nodes repeatedly call a random neighbor to transfer their
messages. At the end of the process, the information should be
spread across the network structure for the algorithm to
converge.
In this paper, based on gossip algorithms we investigate the
spreading of rumors over a social network. We will start by
describing the case of two conflicting messages spreading by
considering an algorithm that has already been proposed in
literature [1]. Given a state of the graph where the two
messages are held by some of the nodes, we analyze the final
state of the graph based on a deterministic model based on the
expectation of a Markov Chain. After the two messages are
spread over the network a consensus could be achieved over
one of the two messages. For that reason, a simple and efficient
consensus-based algorithm is proposed to attain this goal.
One other important aspect of gossip algorithms is their
ability to model human behavior. For instance, in a social
network where the agents are assumed to make rational
decisions these agents base their belief choices on likelihood
criteria. For this reason, we decided to investigate the use of
the consensus-based gossip algorithm in game theory, more
precisely in a voting scenario where multiple agents act
according to their local knowledge. Based on that scenario, we
propose an application of gossip algorithms in modeling the
Word of Mouth (WoM ) marketing strategy where customers
contribute in the marketing strategy of a given product or
service.
The rest of this paper is organized as follows. In Section II,
a protocol modeling the case of two messages being forwarded
over the same network is proposed and analyzed. Then, an
algorithm for achieving consensus over two spread messages is
described in Section III. Section IV provides two different
applications of gossip algorithms, one in game theory and the
other in marketing . Finally, we finish with some conclusions
and future directions in Section VII.
II. T
WO
C
ONFLICTING
R
UMORS
S
PREADING
OVER
THE
S
AME
N
ETWORK
We investigate the problem of spreading two conflicting
messages m1 and m2 on a social network.
For concreteness, let us begin with a review of a message
spreading algorithm. In a social network, represented by a
graph, informed nodes aim at sending their message to their
neighbors. We propose an asynchronous gossip based model.
At each round, a node i is chosen at random over the network.
If i is holding one of the two messages, i selects a neighbor j
uniformly at random and informs it. We point out that this
approach does not give a realistic representation of the problem
since a node may lose interest in spreading the message if it
tries to inform another node already knowing it. To tackle this
issue, we propose to study a different model proposed in [1].
A. Algorithm description :
Let G=(V,E) be a complete graph where |V|=n and E=V
x
V
and let l be a positive integer l IN
. The algorithm proceeds as
follows :
In the beginning, two subsets of V, I
1
and I
2
, are
respectively informed by m1 and m2. At each round :
·
A node i is chosen at random over V. if i holds a
message :
i chooses uniformly at random a neighbor j with a
probability 1/(n-1).
If node j doesn't have the message (susceptible
node), node i informs it (j becomes an m1
infective node). Else, i increments a counter C
i
with one (C
i
counts the unnecessary calls made
by i, their initial value is 0).
If C
i
=l, node i stops spreading the message and
becomes a removed node.
To start, we will assume that l=1. According to the
algorithm description, nodes can be in one of the five following
states:
·
State 1: Has (m1) and is spreading it "m1 infective".

·
State 2: Has (m2) and is spreading it "m2 infective".
·
State 3: Has (m1) and stopped spreading it " m1
removed".
·
State 4: Has (m2) and stopped spreading it "m2
removed".
·
State 5: Doesn't have a message "susceptible"
Following these five states, at each step k let I1(k), I2(k),
S(k), R1(k) and R2(k) denotes number of nodes in state 1, 2, 3,
4 and 5 respectively.
B. Markov Chain Model
If we denote by X(k) the vector (I1(k), I2(k), S(k), R1(k),
R2(k))
T
, the communication step depends only on X(k) the state
of the nodes at k, thus X(k+1) depends only on X(k). The
process can then be modeled with a DTMC (Discrete Time
Markov Chain).
Let us compute its transitions probabilities. Actually, given
a state S(k), there are 5 possible transitions :
·
p
0
(k)=Pr(X(k)(I(k)+1, I2(k), Y(k)-1, R1(k), R2(k)))
p
0
(
k )=
S(k)+R1(k)+R2(k )
N
(1)
·
p
1
+
(k)=Pr(X(k)(I1(k)+1, I2(k), Y(k)-1,R1(k),R2(k)))
p
+(
k)
1
=
I1(k) . S(k)
N ( N-1)
(2)
·
p
1
-
(k)=Pr(X(k)(I1(k)-1, I2(k), Y(k), R1(k)+1,R2(k)))
p
+(
k)
1
=
I1(k) .( N-S( k))
N (N -1)
(3)
·
p
2
+
(k)=Pr(X(k)(I(k), I2(k)+1, Y(k)-1, R1(k), R2(k)))
p
+(
k)
2
=
I2(k ). S( k)
N ( N-1)
(4)
·
p
2
-
(k)=Pr(X(k)(I(k), I2(k)-1, Y(k), R1(k), R2(k)+1))
p
+(
k)
2
=
I2(k ).( N -S(k))
N ( N-1)
(5)
C. Deterministic model
The DTMC introduced above is reducible and transient as
states of the form (0, 0, j, r1) are absorbing states. In order to
have a steady-state distribution, the first DTMC is modified as
follows: Given an initial state (n1, n2, n-n1-n2, 0), and for any
j and r1 = n1, p0, 0, j, r1(n1, n2, N-n1-n2, 0) = 1.
However, the computation of its steady-state has a high
complexity and so we will compute its conditional expectation
and then deduce a deterministic model.
The expectation calculi give :
E [I1( k+ 1)I1(k )]= I1( k)+
I1(k ). (2.S( k)- N )
N ( N-1)
(6)
E[I2 (k+1)I2(k )]= I2(k)+
I2(k ).(2.S (k)- N )
N (N -1)
(7)
E[S(k +1)S(k )]=S( k)-
S(k) .( I1(k)+I2( k))
N ( N-1)
(8)
E[R1(k +1)R1(k)]=R1(k)+
I1(k) .( N-S( k))
N (N -1)
(9)
E[R2( k+1)R2( k)]=R2(k )+
I2(k ).( N -S(k))
N ( N-1)
(10)
Thus, if we denote by i1, i2, s, r1 and r2 respectively
"infective m1", "infective m2", "susceptible", "removed m1"
and "removed m2" nodes,and using (6), (7), (8), (9), (10) and
the two assumptions:
·
N~N-1 for high N
·
Time intervals between the Poisson clock ticks is
neglected (i.e.: k=t).
A deterministic model is deduced :
di1(t)
dt
=
i1(t)(2s(t)-1) (11)
di2 (t)
dt
=
i2 (t)(2s(t)-1) (12)
ds(t )
dt
=-
s( t)(i1(t)+i2 (t)) (13)
dr1 (t)
dt
=
i1 (t)(1-s(t)) (14)
dr2 (t)
dt
=
i2(t )(1- s(t)) (15)
Furthermore, we can note that :
di1(t )
di2 (t)
=
i1(t)
i2(t)
and
dr1(t)
dr2 (t)
=
i1( t)
i2 (t)
(16)
which gives an interesting relation between i1(t) and i2(t), r1(t)
and r2(t) :
i1(t)=
i1( 0)
i1(0)+i2(0)
. i(t) (17)
r1(t)=
i1(0)
i1( 0)+i2 (0)
. r( t) (18)
where i(t)=i1(t)+i2(t) and r(t)=r1(t)+r2(t).
As a consequence, the problem is reduced to one message
spreading over a social network. Then, we reduce (11), (12),
(13), (14) and (15) to two equations. Moreover, in [3], it is
showed that even if l1 equations still hold by adding a
multiplying coefficient 1/l. Let s, i and r denote respectively
the fractions of susceptible, infective, and removed individuals,
such that s + i + r = 1 :
ds(t )
dt
=-
si (19)
di (t)
dt
=
si+
1
l
(
1- s)i (20)
The solution of this couple of equations is [3]:
i(s)= l+1
l
(
1- s)+ 1
l
log( s) (21)
This gives us a first indication on how i changes with s (Figure
1).

The algorithm stops when all the infective nodes stop
spreading the message, i.e: i = 0. According to equation (21),
i(s) is zero when :
s=exp(-(l+1)(1- s)) (22)
Which gives an implicit solution of the equation. Here are
some theoretical results for the reach (number of nodes that
end by having the message) given two different values of k :
- For l=1, s=20%.
- For l=2, s=6%.
When in the final state, we can deduce easily r1 and r2 by
multiplying r by the initial coefficient respectively i1(0)/i(0)
and i2(0)/i(0).
D. Simulation results:
The first simulation ( Figure 1) shows the evolution of i as
a function of s for different l values. We can note that as the
value of l increases the curve gets closer to a linear shape.
This is justified by the fact that (21) becomes i=1-s which is
in fact a linear function. Moreover, the plot of the theoretical
curve fits the simulation results when l=1 which validates the
deterministic model.
Then, using the Monte Carlo method, the simulation
results in Figure 2 were obtained. Figure 2 shows the mean of
the difference between the two messages sets cardinalities as a
function of the initial difference. We can note the linear shape
of the curve as expected with the deterministic model in the
last section.
E. Multiple messages
In the case of multiple messages, we assume that we are
concerned by only one of these messages and the rest are
considered as disrupting messages. Hence, the problem can be
reduced to a two conflicting messages spreading over the same
network (one for the valid message and the other englobing all
the adversary messages). All the results presented in this
section could be applied.
III. C
ONSENSUS
OVER
TWO
DIFFUSED
MESSAGES
We assume that the initial state of the graph is as follows :
- n1 nodes received message (g1).
- n2 nodes received message (g2).
- All the nodes received a message : n1 + n2 = n.
Then, we want all the nodes to agree on one of the
messages. We start by considering an asynchronous algorithm
that uses two increasing counters and we show that it is
equivalent to a seemingly simpler algorithm that only uses one
counter averaged at each step.
First, we assume that each node i holds two counters for
received messages (a counter C
i
for the messages
corresponding to his own message, and a second counter C'
i
for
those corresponding to the other message).
All the counters start by a value of zero. At each step k, a
randomly chosen node i is woken up. Then, a neighbor j is
chosen uniformly at random. Both nodes exchange messages
and counters, then update their counters as follows :
- if the received message is different from the held
message, its counter is incremented by the value of the
received counter (C'
i
(k+1)=C'
j
(k+1)=C'
i
(k)+C
'j
(k)). The new
message is stored.
Fig. 1 Infective nodes ratio as a function of susceptible nodes
ratio (Theoretical result for l= 1 and simulation results for
l=1,2,4,5)
Fig. 2 Mean of the difference between the two messages sets
as a function of the initial difference : comparison between
simulation and theoretical results (n=5000 and n1+n2=200).
- if the received message corresponds to the held message,
the counter corresponding to the held message is incremented
by the value of the received counter (C
j
(k+1)=C
i
(k+1)=C
i
(k)
+C
j
(k)).
For a node i at step k, to choose the most relevant message,
i simply compares his two counters. Comparing the counters is
equivalent to subtracting the value of C'
i
from that of C
i
.
Thus we propose the following algorithm to study. We give
to each node i a counter C
i
and proceed by averaging the
counters to achieve consensus as described below.
A. Algorithm description :
Initialization step :
for i in {1,..,n}
C
i
=
{
1,if iholds( g1)
-
1, if iholds( g2)
}

Communication step :
At the rate of a Poisson process (step k) , for each node i :
- i wakes up.
- i chooses a neighbor j uniformly at random.
- Both nodes update their counters :
C
j
(k+1)=C
i
(k+1)=(1/2)( C
i
(k)+C
j
(k))
- Both i and j sleep.
B. Convergence analysis :
In this section, we will study the convergence of the
randomized gossip algorithm. In the algorithm described above
nodes proceed by updating their local counter at each step. Let
N
i
be the set of i's neighbors, n
i
=|N
i
| the number of i's neighbors
and C(k) the vector for which entries are the counters C
i
(k) at
each time slot k. Thus, the update can be modeled linearly by
the following equation:
C (k +1)=W (k ). C (k ) (23)
where, when node
i chooses node j from N
i
:
W (k )= I -
(
e
i
-
e
j
)(
e
i
-
e
j
)
T
2
(24)
with probability
1
n.n
i
, where e
i
is an n x 1 unit vector with
the i
th
component equal to one.
W (k) must satisfy some constraints according to [2]. These
constraints are imposed by the gossip algorithm and the graph
topology.
If nodes i and j are not neighbors, W
ij
(k) must be zero.
Further, since every node can communicate with only one of its
neighbors per time slot, each column of W (k) can have only
one non-zero entry other than the diagonal entry.
In each iteration, the averaging computation impose the
preservation of the sums : this means that 1
T
.W (k) = 1
T
, where
1 denotes the vector of all ones. Also, the vector of averages
must be a fixed point of the iteration, i.e. W (k).1 = 1.
Since the choice of i and j are independent of the time slot,
W(k) matrices are IID. Secondly, from (23), we can find by
iteration that :
C( k)=(
l=0
l=k-1
W (l)). C( 0)= P(k-1). C(0) (25)
Hence, if C(k) must converge to C
ave
.1 we must have :
lim
k
P(k )=1.1
T
n
(26)
In the next sections, the convergence to the initial counters
mean will be proven.
1) Convergence in expectation:
Let W = E[W(k)]. Under the following conditions :
(a1)- 1
T
.W = 1
T
(a2)- W .1 = 1
(a3)- (W - 1.1
T
n
)
1 where (.) is the spectral
radius of a matrix.
We have :
lim
k
E( P( k))= 1.1
T
n
(27)
Then: lim
k
E( P( k))=
i=1
i=n
C
i
(
0)
n
(28)
2) Convergence of the second moment :
The convergence of the second moment is investigated in
this section to quantify the convergence rate of C(k) to C
ave
.
To obtain it, lets consider the error : N(k)=C(k) ­ C
ave
.1.
Considering the evolution of N(k), we can easily demonstrate
that :
N (k+1)=W (k) . N (k ) (29)
So, N(k) evolves with the same linear system as C(t). Hence
we can write :
E [N ( k+1)
T
. N ( k +1)N ( k)] =
N ( k )
T.
E [W (k)
T
. W ( k)]. N ( k) (30)
Using the fact that W(k) is doubly stochastic and so is W,
and the orthogonality between N(t) and 1, we can demonstrate
that :
E[N ( k)
T
N ( k)]
2
2t
(
E[W
TW
])
N (0) (31)
Hence, since
2
1 (second largest eigen value of W ), the
expectation of the error converge to zero when k approaches
infinity.
3) High probability bounds on averaging time :
In [2], an upper and lower bounds are demonstrated.
Theorem 1 :[2] Having a gossip algorithms with an initial
state C(0):
For k 6.K*()
: Pr (
C ( k)-C
ave
C (0)
) (32)
For k K*() : Pr (
C ( k)-C
ave
C (0)
) (33)
Where : K
.
.
(
)=
log(
-
1
)
2.log(
2
(
W )
-
1
)
(34)
These are results for averaging consensus. However, the
consensus studied her is to reach all the nodes to have the same
decision on the message they spread. It's clearly a consensus as
it's a fixed point of the algorithm.
To reach that, a sufficient but not necessary condition can
be implemented as a stopping criterion :
C (k)-C
ave
C
ave
(35)
This means that :
1in ,
C
i
(
k )-C
ave
C
ave
(36)
Hence, with (4), we can obtain the corresponding for the
convergence. For =
C
ave
C (o)
=
i=0
n
C
i
(
0)
n
(
n)
=
n
1
-
n
2
n
(
n )
(37)
Note that
is an increasing linear function of the initial
difference between the number of g1 and g2 holders. We have:
Pr (C
i
(
k)have the same sign)1-
n
1
-
n
2
n
(
n)
(38)

for every k
3.log(
C ( o)
C
ave
)
log(
2
(
W )
-
1
)
(39)
and :
Pr ( All C
i
(
k) have the same sign)1-
n
1
-
n
2
n
(
n)
(40)
for every k
log(
C (o)
C
ave
)
2.log (
2
(
W )
-
1
)
(41)
Which gives the consensus reach bounds of the algorithm.
C. Simulation results
First, we simulate (Figure 3) the evolution of the number of
nodes holding each one of the two messages (g1) and (g2).The
two simulations concern a complete graph with 1000 nodes.
This simulation shows that the algorithm converges and the
nodes end up agreeing on the most relevant message (i.e: the
one with the higher initial number of holders).
Secondly, the curves in Figure 4 show the evolution of the
distance
C (k)-C
ave
. Foremost, we can see that the
convergence is reached even before getting a distance of zero
between the counter's vector and C
ave
, which can be justified by
(26). In fact, the convergence is a convergence of the signs of
the counters rather than the counters themselves. Then, it is
also clear ( Figure 4) that when the initial settings of n1 and n2
are farther from each other the convergence rate is higher,
which matches the theoretical results.
IV. A
PPLICATION
A. A repeated game of distributed voting:
We consider a repeated game where players can vote for
one of two possible candidates. Each agent prefers one of the
two candidates over the other. However, a player is better off
voting for the candidate that is going to win anyway. Hence,
we define u
k
(i), the payoff of each player i at a step k, as
follows:
u
k
(
i)=( Sum of all the agents agreeing with i)
Thus, the set of agents N is partitioned so that we have two
subsets N
1
and N
2
containing respectively the agents agreeing
over the choice
1
or
2
.
Since the agents are unaware of the choices of other agents
they cannot know what their payoffs are at a certain step.
Moreover, we assume that at each step, to agents are able to
contact each other. The goal is to find a strategy that will help
all the agents maximize their payoffs.
Proceeding as shown in the algorithm provided in Section
III. All the players hold a counter that is initialized at the
beginning of the game.
At a step k, we assume that no agent is able to observe his
payoff. Instead, each couple of agents i and j who came in
contact with each other can observe the state of their counters.
According to the results of the previous section, consensus
over the choice of a candidate is reached with high
probability.
Fig. 3 Evolution of holders of each message (g1: lower curve,
g2: upper curve) at each round (initial values: n1=400 and
n2=600)
Fig. 4 Distance to the expected average at each round k(i.e.:
C (k)-C
ave
). Three initial settings are distinguished
B. Application to word-of-mouth marketing
Direct marketing deals with separate events: each email or
advertisement is considered as a separate deal. But in a
community where people are related to each other, all of these
notions are connected.
We analyze the case where two conflicting products are
spreading in a community. There aren't many distinct
thresholds for spreading two products in a community: one of
the products will win at the very end. But in general, the
spreading of the products stops before this state is achieved.
The individuals proceed as shown in the algorithm
described in Section III. We start by considering that all the
nodes (individuals) are connected to each other. If the graph is
not complete, the counters will be weighted proportionally to
the centrality of the nodes. We can choose betweenness as a
measure of centrality. The intuition behind this approach is
that a node by which transits more information is more likely
to have an impact on the other nodes. The only thing that
changes is the initialization step and we are brought back to

the complete graph structure. Thus, we consider a set of agents
V connected in a complete undirected graph G = (V, E).
Instead of initializing all the nodes at the same value of the
counter, a higher value should be assigned to the counters of
the nodes that are more convincing.
First, we assume that more or less convincing nodes are
distributed according to a normal distribution. In other words,
in the initialization step: the random variable that assigns a
value to a counter follows a Gaussian distribution.
According to Section III, at each step, each counter is a
linear combination of the initial values of the counters.
Consequently, at each step, the vector C(k) follows a normal
distribution. Moreover, the values of the counters converge to
the mean of the initial distribution. Hence, the mean of the
final distribution is equal to the initial mean. However, the
closer this initial mean is to zero, the slower is the
convergence of the algorithm.
Simulation results:
Experimental results (figure 5) show that the final
distribution is as expected a normal distribution.
Changing the initial variance of the distribution only
impacts the convergence rate.
We simulate a graph where the distribution has a mean that
has a negative value close to zero. As shown in Figure 5, the
individuals end up having the same preference, i.e.: all the
counters are negative.
However, in the real world the spreading process stops
before convergence is reached since new products come out.
Fig. 5: Last distribution of the counters (the values are
condensed around a value close to -0.0112)
V. C
ONCLUSION
AND
F
UTURE
W
ORK
Gossip algorithms are an efficient tool to model rumor
spreading over network structures. The results presented in this
paper give an overview of the way rumor forwarding proceeds
in a social network.
In the case of multiple conflicting messages, our main
result is that the number of holders of a given message evolves
almost proportionally to the sum of all the messages holders
with a coefficient that is equal to the initial number of holders.
Furthermore, the consensus based gossip algorithm makes
it possible to lead the nodes towards choosing the same belief.
Of course, this is does not represent realistic scenarios as many
parameters were not considered. For instance, there might exist
some non-cooperating agents among the nodes.
A game theoretical approach shows that agents can
maximize their payoffs based only on their local knowledge
and agree on the same choice in a voting game.
A word-of-mouth marketing model shows how, in a
community where two conflicting ideas start spreading, only
one idea remains in the end.
There are more issues to be explored. In multiple rumor
spreading the consideration of other types of graphs may lead
to important practical results. So far we have avoided the
impact of the position of nodes, their study could optimize the
spreading by getting through strategical nodes. In the
consensus-based approach more work could be realized on the
influence of weighted graphs on the convergence of the
algorithm. In other words, the updating step could be realized
with multiplying the counters by a factor that illustrates the
influence of each one of the two communicating nodes.
VI. A
CKNOWLEDGMENT
We would like to thank Vincent Gripon and Michael
Rabbat for their supprot and supervision. We would also like
to thank Samir Saoudi for his time and consideration. Finally,
many thanks to the host laboratory in the electronics
departement at Télécom Bretagne.
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6 of 6 pages

Details

Title
Algorithm for Achieving Consensus over Conflicting Rumours
Grade
bac+4
Authors
Year
2014
Pages
6
Catalog Number
V278179
ISBN (Book)
9783656713586
File size
640 KB
Language
English
Tags
algorithm, consensus, rumours, gossip algorithms, social networks, game theory, word-of-mouth.
Quote paper
Ismail Elouafiq (Author)Amine Semma (Author), 2014, Algorithm for Achieving Consensus over Conflicting Rumours, Munich, GRIN Verlag, https://www.grin.com/document/278179

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